4442 Particle Physics Ryan Nichol Module 3 The Dirac Equation - - PowerPoint PPT Presentation

4442 Particle Physics Ryan Nichol Module 3

The Dirac Equation

• Time to move on from toy theories
• Need a proper theoretical description

– Spin 1/2 matter particles (Fermions) – Interacting via spin 1 force particles (Bosons) – Must be relativistic

• Essentially need to find a replacement for the non-

relativistic Schrödinger equation which originates from:

• using the standard quantum mechanics operators for

energy and momentum

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p2 2m + V = E

i@Ψ @t = ˆ HΨ = 1 2mr2Ψ + V Ψ ˆ p = i~ r ˆ E = +i @ @t

ˆ Pµ = i@µ

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The Dirac Equation

• The Klein-Gordon equation
• is a relativistic : E2 = p2 + m2 : wave equation but there is a

troublesome negative probability density solution (arising from ).

• Dirac managed to produce an equation valid in both the relativistic

and non-relativistic regimes which also has a “natural” interpretation of the negative energy solutions - he invented the notion of anti-particles (with positive probability density solutions) which were then experimentally verified a year later.

• Derivation of Dirac equation rests on the introduction of 2x2

(Dirac)-matrices since matrices can have the property that AB + BA = 0 whereas numbers are commutative and this allowed : positive density solutions since the equation contained no E2 (-∂2ψ/ ∂t2) terms and for the solutions to satisfy E2 = p2 + m2 .

@µ@µΨ + m2Ψ = 0

E = p p2 + m2

Dirac gamma Matrices satisfy : The Dirac equation is written as : The Dirac Hamiltonian is (derivation in problem sheet)

• r

And

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Derivation of the Dirac Equation

Dirac Gamma Matrices

• Pauli spin matrices are 2x2 matrices
• Dirac gamma matrices are 4x4 matrices
• e.g

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p σ1 = ✓0 1 1 ◆ , σ2 = ✓0 −i i ◆ , σ3 = ✓1 −1 ◆ γ0 = ✓ 1 −1 ◆ , γi = ✓ σi −σi ◆ , γ5 = ✓ 1 1 ◆

γ2 = B B @ −i i i −i 1 C C A

Useful Operators

• Dirac Hamiltonian
• Helicity
• Spin
• Total angular momentum

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B @ − C A ˆ H = ✓ m ~ · ~ p ~ · ~ p −m ◆

ˆ h = 1 2 ✓~ · ˆ p ~ · ˆ p ◆

✓ · ~ Σ = ✓ ~

• ~
• ◆

✓ ◆ ~ J = ~ L + 1 2 ~ Σ ~ S = 1 2 ~ Σ

Solutions for particles at rest

• Dirac equation:
• For a particle at rest:
• Therefore:
• And:

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(µ@µ − m) Ψ = 0

@Ψ @x = @Ψ @y = @Ψ @z = 0

~ p = 0

i0 @Ψ @t − mΨ = 0

✓ 1 −1 ◆ ✓ ∂ΨA

∂t ∂ΨB ∂t

◆ = −im ✓ ΨA ΨB ◆

ΨA = ✓ Ψ1 Ψ2 ◆ ΨB = ✓ Ψ3 Ψ4 ◆

✓ ◆ @ΨA @t = −imΨA

@ΨB @t = +imΨB ΨA(t) = ΨA(0)e−imt ΨB(t) = ΨB(0)e+imt

And can be expressed in terms of 2-spinors: χ1 , χ2 The solutions of the Dirac equation : The “anti-particle” solutions : are known as Dirac spinors. There are four independent solutions to the Dirac equation: two “particle” and two “anti-particle” solutions. Each describes (anti-)particles with mass and spin ½ e.g. quarks and electrons, in two helicity states. Derivation of solutions **

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Solutions

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Interpretation

• Dirac’s Interpretation of negative energy solutions
• If the -ve energy solution where an electron then we could get the situation

where a +ve energy electron could interact with a photon to reduce it’s energy to be the more naturally preferred (and allowable by the Dirac eqn) -ve energy and this is not observed i.e. the negative energy solutions cannot be electrons.

• Vacuum is a sea of -ve energy electrons. If we create a “hole” in the sea e.g.

by hitting one of the -ve E electrons with a photon of E > 2me then the sea effectively increases in charge by +1, and becomes more positive in energy -

• ur “hole” thus looks like a +ve E particle of +ve charge with the mass of an
• electron. This hole can be identified as a real particle - the positron with
• pposite charge, momentum, spin and energy to the original -ve E electron

eigenstate but the same mass.

• Shares many similarities with the discussion of holes in “Fermi seas” of

semiconductors in condensed matter physics.

• Discussion is actually more complex since we need to introduce Quantum

Field Theory operators for particle creation/annihilation……

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Next Module

• The continuity equation
• “interaction currents” in terms of gamma matrices
• Lagrangian formalism
• The addition of EM interactions (Maxwell’s equations)

into the Dirac equation i.e. a real theory

– Quantum ElectroDynamics (QED)